Appendix I Processes: phyto biomass rates

The rates that determine the temporal evolution of a phytoplankton group’s biomass are primary productivity, respiration, exudation and settling. In general terms these processes can be thought of as follows:

• Primary productivity is the consumption of carbon and the generation of oxygen and carbohydrate due to photosynthesis. Photosynthetically active radiation catalyses this process so it occurs only during daylight hours. This process is also referred to as growth or gross primary productivity
• Respiration is the consumption of oxygen and stored carbohydrate and the production of carbon. This can be thought of as the reverse of primary productivity. Respiration dominates phytoplankton dynamics during nighttime hours, although it generally still operates during the daytime
• Exudation (also referred to as photo exudation) is the loss of phytoplankton biomass as a result of primary productivity
• Settling is the physical settling of phytoplankton, including motility

The general form of the phytoplankton flux relationship for any given computational cell is therefore given by Equation (I.1).

\[\begin{equation} F^{phy} = \text{Primary productivity} - f(\text{Respiration, Exudation}) - \text{Settling} \tag{I.1} \end{equation}\]

\(f\) represents functions described in Appendix L. The fundamental rates in Equation (I.1) - primary productivity, respiration, exudation and settling - are described following.

I.1 Primary productivity

The primary productivity rate is computed in a series of stages within the WQ Module, where the combination of these stages used for any given phytoplankton group depends on group configuration. These stages are described below. For clarity, supporting functions nested within primary productivity rate calculations (such as limitation functions) are initially presented in passing, but are cross referenced to detailed descriptions in subsequent Appendices.

The rate of primary productivity (often referred to as just productivity or growth rate) of a phytoplankton group is a /day rate. It is computed initially within the WQ Module as per Equation (I.2).

\[\begin{equation} R_{prod\langle computed\rangle}^{phy} = R_{prod}^{phy} \times L_T^{phy} \times \text{min}\left(L_{lght}^{phy},L_{nit}^{phy},L_{phs}^{phy},L_{sil}^{phy}\right) \tag{I.2} \end{equation}\]

For each phytoplankton group, \(R_{prod}^{phy}\) is the user specified (or default) phytoplankton productivity (growth) rate at 20\(^o\)C with no light, temperature, salinity, silicate or nutrient limitation, and \(L_T^{phy}\), \(L_{lght}^{phy}\), \(L_{nit}^{phy}\), \(L_{phs}^{phy}\), \(L_{sil}^{phy}\) are the computed limitation functions for water temperature, light, nitrogen, phosphorus and silicate, respectively. These functions, when appropriate, act to modify \(R_{prod}^{phy}\), noting that only \(L_T^{phy}\) always does so. The other limitation functions in Equation (I.2) act collectively, with only the most limiting (i.e. the smallest function value) of \(L_{lght}^{phy}\), \(L_{nit}^{phy}\), \(L_{phs}^{phy}\) and \(L_{sil}^{phy}\) being selected to also multiplicatively modify \(R_{prod}^{phy}\) with \(L_T^{phy}\).

If nitrogen fixing is implemented, then the productivity rate computed in Equation (I.2) is further modified as per Equation (I.3).

\[\begin{equation} R_{prod\langle computed\rangle}^{phy} = R_{prod\langle computed\rangle}^{phy} \times \left(f_{nfix}^{phy} + L_{nit}^{phy} \times \left(1.0 - f_{nfix}^{phy}\right)\right) \tag{I.3} \end{equation}\]

\(f_{nfix}^{phy}\) accounts for the metabolic cost of nitrogen fixing on primary productivity and is the user specified (or default) reductive factor applied to productivity under full nitrogen fixing conditions. Specifically:

• Full nitrogen fixing occurs when \(L_{nit}^{phy}\) is zero (i.e. no water column nitrogen is available for uptake), and Equation (I.3) then has \(R_{prod\langle computed\rangle}^{phy}\) reduced by the factor \(f_{nfix}^{phy}\)
• No nitrogen fixing occurs when \(L_{nit}^{phy}\) is one (i.e. excess water column nitrogen is available for uptake), and Equation (I.3) then has \(R_{prod\langle computed\rangle}^{phy}\) unchanged

Salinity limitation of primary productivity is not mandatory, and can be switched on and off by the user, or equally set to be an enhancer of respiration rather then reducer of productivity. If salinity limitation of productivity is implemented, then the \(R_{prod\langle computed\rangle}^{phy}\) computed above is further modified as per Equation (I.4).

\[\begin{equation} R_{prod\langle computed\rangle}^{phy} = R_{prod\langle computed\rangle}^{phy} \times L_{sal-pp}^{phy} \tag{I.4} \end{equation}\] \(L_{sal-pp}^{phy}\) is the limitation on primary productivity due to salinity and is between zero and one, as per the other limitation functions noted above.

Once the rate of primary productivity has been computed by applying Equation (I.2), and potentially Equations (I.3) and (I.4), the corresponding flux of carbon (i.e. phytoplankton growth) to a phytoplankton group is computed as per Equation (I.5).

\[\begin{equation} \href{AppDiags.html#WQDiagPhyPriProd}{F_{prod\langle computed\rangle}^{phy}} = R_{prod\langle computed\rangle}^{phy} \times \left[PHY\right] \tag{I.5} \end{equation}\] \(\left[PHY\right]\) is the ambient phytoplankton group concentration. These fluxes are summed over all simulated phytoplankton groups to compute the community primary productivity, \(F_{prod\langle computed\rangle}^{comm}\).

The corresponding fluxes of nitrogen due to this primary productivity are described by Equations (K.2) (basic model) and (K.4) (advanced model). Similarly, the corresponding fluxes of phosphorus are described by Equations (K.7) (basic model) and (K.8) (advanced model). The same diagnostic variable name is used to report these fluxes, regardless of the phytoplankton model used.

The WQ Module uses a minimum concentration \(\left[PHY\right]_{min}\) to maintain phytoplankton concentrations above zero. This is because if concentrations decrease to zero, then by Equation (I.5), phytoplankton fluxes (and therefore concentrations) will remain at zero regardless of computed primary productivity rates.

I.2 Respiration

The respiration rate is computed in a series of stages within the WQ Module, where the combination of these stages used for any given phytoplankton group depends on group configuration. These stages are described below. For clarity, supporting functions nested within respiration rate calculations (such as limitation functions) are initially presented in passing, but are cross referenced to detailed descriptions in subsequent sections.

The rate of respiration of a phytoplankton group is a /day rate. It is computed initially within the WQ Module as per Equation (I.6).

\[\begin{equation} R_{resp\langle computed\rangle}^{phy} = R_{resp}^{phy} \times \underbrace{\left[\theta_{resp}^{phy}\right]^{\left(T-20\right)}}_{\text{Influence of temperature}} \tag{I.6} \end{equation}\]

\(R_{resp}^{phy}\) is the user specified phytoplankton respiration rate at 20\(^o\)C, \(\theta_{resp}^{phy}\) is the temperature coefficient for respiration, and \(T\) is ambient water temperature. Figure I.1 presents the influence of \(\theta_{resp}^{phy}\) on \(R_{resp}^{phy}\), as a function of temperature.

Figure I.1: Move the slider to see the effect of changing temperature coefficient on the computed phytoplankton respiration rate

As for primary productivity, salinity limitation of respiration is not mandatory, and can be switched on and off by the user. If salinity limitation of respiration is implemented, then the \(R_{resp\langle computed\rangle}^{phy}\) computed above is further modified as per Equation (I.7).

\[\begin{equation} R_{resp\langle computed\rangle}^{phy} = R_{resp\langle computed\rangle}^{phy} \times L_{sal-r}^{phy} \tag{I.7} \end{equation}\]

\(L^{phy}_{sal-r}\) is the limitation function on respiration due to salinity, and is always greater than one. This limitation function can therefore be conceptualised as a respiration enhancer: it acts to reduce biomass via increasing respiration rather than reducing primary productivity. Salinity limitation is applied to only one of primary production (Section I.1) or respiration, depending on the salinity limitation model selected, and not both.

If a group’s ambient phytoplankton concentration is less than the user defined (or default) minimum concentration \(\left[PHY\right]_{min}\), then the respiration rate is set to zero.

Once the respiration rate has been computed, the corresponding flux of carbon (i.e. phytoplankton respiration) from a phytoplankton group due only to respiration (i.e. excluding mortality and excretive losses) is as per Equation (I.8).

\[\begin{equation} \href{AppDiags.html#WQDiagPhyResp}{F_{resp\langle computed\rangle}^{phy}} = R_{resp\langle computed\rangle}^{phy} \times f_{true-resp}^{phy} \times \left[PHY\right] \tag{I.8} \end{equation}\] \(f_{true-resp}^{phy}\) is the user specified (or default) fraction of respiration that corresponds to the generation of energy via consumption of stored chlorophyll a (i.e. the fraction that does not result in excretive or mortality losses) and \(\left[PHY\right]\) is the ambient phytoplankton group concentration. For a given phytoplankton group, net productivity is calculated as primary productivity, less the flux computed in Equation (I.8) (i.e. respiration due only to energy generation). These net productivity fluxes are summed over the number of simulated phytoplankton groups (i.e. n = 1 to num_phy) to compute the community net primary productivity as per Equation (I.9).

\[\begin{equation} \href{AppDiags.html#WQDiagCPhyNetProd}{F_{netprod\langle computed\rangle}^{comm}} = \sum_{n=1}^{\text{num_phy}} \left[ F_{prod\langle computed\rangle}^{phy_n} - F_{resp\langle computed\rangle}^{phy_n} \right] \tag{I.9} \end{equation}\]

The respiration fluxes of nitrogen and phosphorus corresponding to Equation (I.8) are as per Equations (I.10) and (I.11), respectively.

\[\begin{equation} \href{AppDiags.html#WQDiagPhyRespN}{F_{resp-N\langle computed\rangle}^{phy}} = R_{resp\langle computed\rangle}^{phy} \times f_{true-resp}^{phy} \times \left[IN\right] \tag{I.10} \end{equation}\]

\[\begin{equation} \href{AppDiags.html#WQDiagPhyRespP}{F_{resp-P\langle computed\rangle}^{phy}} = R_{resp\langle computed\rangle}^{phy} \times f_{true-resp}^{phy} \times \left[IP\right] \tag{I.11} \end{equation}\]

\(\left[IN\right]\) and \(\left[IP\right]\) are the internal phytoplankton nitrogen and phosphorus concentrations, respectively. These are either constant ratios of nitrogen and phosphorus to phytoplankton biomass (i.e. the basic phytoplankton model) or computed variables (i.e. the advanced phytoplankton model), and the WQ Module will use the appropriate value based on the phytoplankton model deployed, on a group by group basis.

I.3 Exudation

The exudation rate of a phytoplankton group is a /day rate. It is computed within the WQ Module as per Equation (I.12).

\[\begin{equation} R_{exud\langle computed\rangle}^{phy} = R_{prod\langle computed\rangle}^{phy} \times f_{exud}^{phy} \tag{I.12} \end{equation}\]

\(R_{prod\langle computed\rangle}^{phy}\) is the previously computed phytoplankton primary production rate (Equations (I.2), and potentially (I.3) and (I.4)), and \(f_{exud}^{phy}\) is the user defined (or default) fraction of primary production lost to exudation. This exudation loss is a linear function.

If a group’s ambient phytoplankton concentration is less than the user defined (or default) minimum concentration \(\left[PHY\right]_{min}\), then the exudation rate is set to zero.

I.4 Settling

The settling rate \(V_{settle}^{phy}\) (also referred to as motility) of a phytoplankton group is applied within the WQ Module for each phytoplankton group, prior to the corresponding primary production, respiration and exudation rate calculations. There are several phytoplankton settling models available within the WQ Module. These are described following, and once the settling rate has been computed, the flux (loss) of phytoplankton carbon to the sediments (i.e. from the bottom cell of each computational water column only) is computed for each phytoplankton group via Equation (I.13).

\[\begin{equation} \href{AppDiags.html#WQDiagPhySedmtn}{F_{sedmtn\langle computed\rangle}^{phy}} = \frac{V_{settle}^{phy}}{dz} \times \left[PHY\right] \tag{I.13} \end{equation}\] \(dz\) is the relevant cell thickness, and divides the flux to produce a per volume result for consistency with other corresponding diagnostics (such as mortality). These fluxes are summed over all simulated phytoplankton groups to compute the community sedimentation rate, \(F_{sedmtn\langle computed\rangle}^{comm}\). The latter is in per area units.

The corresponding sedimentation flux of phytoplanktonic nitrogen is computed via either Equation (I.14) (basic model) or (I.15) (advanced model). The same diagnostic variable name is used to report these fluxes, regardless of the phytoplankton model used.

\[\begin{equation} \href{AppDiags.html#WQDiagPhySedmtnN}{F_{sedmtn-N\langle computed\rangle}^{phy}} = \frac{V_{settle}^{phy}}{dz} \times \left[PHY\right] \times X_{N-C-con}^{phy} \tag{I.14} \end{equation}\] \(X_{N-C-con}^{phy}\) is the specified (or default) constant ratio of internal nitrogen to carbon in the phytoplankton group being considered.

\[\begin{equation} \href{AppDiags.html#WQDiagPhySedmtnN}{F_{sedmtn-N\langle computed\rangle}^{phy}} = \frac{V_{settle}^{phy}}{dz} \times \left[IN\right] \tag{I.15} \end{equation}\] \(\left[IN\right]\) is the internal phytoplankton nitrogen concentration.

The corresponding sedimentation flux of phytoplanktonic phosphorus is computed via either Equation (I.16) (basic model) or (I.17) (advanced model). The same diagnostic variable name is used to report these fluxes, regardless of the phytoplankton model used.

\[\begin{equation} \href{AppDiags.html#WQDiagPhySedmtnP}{F_{sedmtn-P\langle computed\rangle}^{phy}} = \frac{V_{settle}^{phy}}{dz} \times \left[PHY\right] \times X_{P-C-con}^{phy} \tag{I.16} \end{equation}\] \(X_{P-C-con}^{phy}\) is the specified (or default) constant ratio of internal phosphorus to carbon in the phytoplankton group being considered.

\[\begin{equation} \href{AppDiags.html#WQDiagPhySedmtnP}{F_{sedmtn-P\langle computed\rangle}^{phy}} = \frac{V_{settle}^{phy}}{dz} \times \left[IP\right] \tag{I.17} \end{equation}\] \(\left[IP\right]\) is the internal phytoplankton phosphorus concentration.

I.4.1 None

In this model, phytoplankton settling \(V_{settle}^{phy}\) is set to zero and phytoplankton are simply advected by the hydrodynamic flow field.

I.4.2 Constant

In this model, phytoplankton settling \(V_{settle}^{phy}\) is set to a constant value, and phytoplankton are settled at this velocity. A negative specification of this quantity corresponds to a downwards settling velocity.

I.4.3 Constant with density correction

In this model, phytoplankton settling \(V_{settle}^{phy}\) is set to a constant value, and phytoplankton are settled at this velocity, but corrected for ambient water density effects, as per Equation (I.18).

\[\begin{equation} V_{sett\langle computed\rangle}^{phy} = V_{settle}^{phy} \times \frac{\mu_{20}\times\rho_w}{\mu\times\rho_{w20}} \tag{I.18} \end{equation}\]

\(V_{settle}^{phy}\) is the specified settling rate (velocity) at 20\(^o\)C, \(\mu\) and \(\rho_w\) are the ambient water dynamic viscosity (in Ns/m\(^2\)) and density (in kg/m\(^3\)), respectively, and \(\mu_{20}\) and \(\rho_{w20}\) are the dynamic viscosity and density of freshwater at 20\(^o\)C, respectively. A negative specification of \(V_{settle}^{phy}\) corresponds to a downwards settling velocity.

I.4.4 Stokes

In this model, phytoplankton settling is computed using the Stokes equation and cell diameter and density, as per Equation (I.19).

\[\begin{equation} V_{sett\langle computed\rangle}^{phy} = -g \times d_{phy}^2 \times \frac{\left(\rho_{phy}-\rho_w\right)}{18\mu} \tag{I.19} \end{equation}\] \(g\) is acceleration due to gravity, \(d_{phy}\) and \(\rho_{phy}\) are phytoplankton cell diameter and density, respectively, and \(\rho_w\) and \(\mu\) are the ambient water density and dynamic viscosity, respectively. This model is only available if phytoplankton cell density is dynamically simulated. Cell density is therefore required to be simulated as a computed variable in both cases, with supporting initial and boundary condition specifications. Cell diameter is fixed at 1 \(\times\) 10 \(^{-5}\) m.

Even though a half saturation light intensity \(I_K\) is not required to compute the settling velocity directly, it is required to compute the cell density term in Equation (I.19). Cell density is computed as described in Appendix M.2. \(I_K\) can therefore be specified in the command to set stokes settling (referred to as \(I_{K-sto}\)), if it is not already specified in a phytoplankton constituent model light limitation function.

I.4.5 Motile

The motility settling model can only be used if internal nutrients are simulated, via deployment of the advanced phytoplankton constituent model.

In this model, phytoplankton are permitted to be motile, with the motility velocity being either a user specified value (and either upwards or downwards), or zero. This motility behaviour is governed by phytoplankton internal nutrient status and ambient environmental conditions, as follows:

• If \(Q \lt 0.675 \times X_{N-C-max}^{phy}\) then phytoplankton is starved of nutrients and swims downwards at a rate \(V_{mot}^{phy}\)
• If \(Q \gt 0.750 \times X_{N-C-max}^{phy}\) then phytoplankton is replete of nutrients and:
  • If local photosynthetically available radiation is greater than \(I_{K-mot}\) then phytoplankton swim up at \(V_{mot}^{phy}\)
  • If local photosynthetically available radiation is less than \(I_{K-mot}\) then phytoplankton are not motile
• If \(0.675 \times X_{N-C-max}^{phy} \lt Q \lt 0.750 \times X_{N-C-max}^{phy}\) then phytoplankton are not motile

\(V_{mot}^{phy}\) is the user specified motility velocity where a negative value is treated as a downwards motility, \(Q\) is the instantaneous ratio of phytoplankton internal nitrogen to carbon computed by the WQ Module (N/C, not \(R_{IN-IP}^{phy}\)), \(X_{N-C-max}^{phy}\) is the maximum internal nitrogen to carbon concentration ratio either user specified (or default) in the advanced constituent model and \(I_{K-mot}\) is the half saturation constant for light limitation of growth. A value of \(I_{K-mot}\) specified in the settling model will be ignored if it has been specified in the light limitation model.

Overriding all of the above is the condition that if local photosynthetically available radiation is greater than 95% of the water surface value, motility is set to zero as phytoplankton are considered to already be at the surface of the model domain.